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Date: 9-9-2019
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Newton's iteration is an algorithm for computing the square root of a number via the recurrence equation
(1) |
where . This recurrence converges quadratically as .
Newton's iteration is simply an application of Newton's method for solving the equation
(2) |
For example, when applied numerically, the first few iterations to Pythagoras's constant are 1, 1.5, 1.41667, 1.41422, 1.41421, ....
The first few approximants , , ... to are given by
(3) |
These can be given by the analytic formula
(4) |
|||
(5) |
These can be derived by noting that the recurrence can be written as
(6) |
which has the clever closed-form solution
(7) |
Solving for then gives the solution derived above.
The following table summarizes the first few convergents for small positive integer
OEIS | , , ... | |
1 | 1, 1, 1, 1, 1, 1, 1, 1, ... | |
2 | A001601/A051009 | 1, 3/2, 17/12, 577/408, 665857/470832, ... |
3 | A002812/A071579 | 1, 2, 7/4, 97/56, 18817/10864, 708158977/408855776, ... |
REFERENCES:
Sloane, N. J. A. Sequences A001601/M3042, A002812/M1817, A051009, A071579 in "The On-Line Encyclopedia of Integer Sequences."
Wolfram, S. A New Kind of Science. Champaign, IL: Wolfram Media, p. 913, 2002.
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